DIGITAL INVESTI3kPIoN O? SWIrCL-IING TRANSIENT F1.OM N IMPEDANCE

SFWNrED CIhCUI2-REAKEH

by

PETEB FiARVEY

A

subnitted to

OhEGON SI1ATE UNIVERSITY

in partial fulfillment of the requirements for the

iegree of

MA2T!.R OF SCIENCE

June 1962

APPROVED:

Redacted for privacy Protssor of1eot1ca1 Enginrtr1g Depart

In Charge of M8jor

Redacted for privacy of t1ectric81 Eng1iiitng Departnnt

Redacted for privacy Chatrman of 'chool Graduate Coaimittee

Redacted for privacy Dean of Grduaiihoo1

Dt' thesis is presentes November 1 1961

Typed by hei1a Hsrvey

AC KI O LEDJEMENT

The author wishes to express rais ratitue

to Professor J. F. Engle fr his irispirtiori,

suggestions and guidance.

TABLE OF CONTENTS

Page

I1'ITRODUCTION. ....... . . . . . . . . . . . . . . . . . . . . . . . 11. 3.

METHOD OF INVESTIGATION . .......... . ......... . . . .......

RESULTS.... . . . . . . . . . . . * . .......... . . . . ....... . . . . . 12

CONCLUSIONS........... ....... ......................... 20

211

APPENDIX... .... .. .. . .. .. ..... ... . ... .. .. . .. . ... ..... .. 25

F 1G URE

ND. Pa e

1. Circuit wtth simulated clrcult-oreaker ...... .... 5

2. 9

j. Ca1cu1at.e. recDvery v1tige curves Casesa t: e ....................................

Li.. Ca1cu1ate recovery voltae curves Casesf to k.. .................................. 15

LIGIrAL INV TIQ1TI;,N OF SWI CHINJ raANsILir$ FßOM N IMPD.NCE

SHUNI1ED C IECUIr-BEEAKfl

INT1ODUC2ION

rhe theret1oi1 cper8tiDn of an 1trnating current

c1rcutt-breicr is bEsically quite simple. At a ranziorn

time Iii the cycle the cnt:ct points sep8rte. rhe

lniucttve elements in tha circuit will sustain an rc

between the cnscts until trie current reaches a zero

int in trie cycle. The arc is trien extinguished and the

ap Detween trie contact points is insulated a&ainst the

resultant rise in volte by the frced entry of a

dielectric medivaì - usually oil r air.

Crie chief cmpiication is that the arc creates a

c-nductin ionizs3 path tnrouh the dielectric medium and

no matter how inienious the desin of the breaker, it is

difficult to reiove all the arc products at each inter-

ruption. If tne separtiDn of the contact points occurs

just before current zero, tnere will be few arc products

but the paints wi'l still be relatively close together at

the m..tural current zero. Conversely, if there is large

separation, tnere will be a lre accuultion of arc products.

jowatsm (1, p.i) has shown tuat there is an

optimum time in the current cycle to initiate trie break,

in Drier to obtain a balance between trie separation DI the

points smi the arc roducts left in trie gap. ie also

2

found the rate of eontct seprit1on t be subor1rute to

the rute of rise of recovery voltage, wnicn is perhaps the

most critical factor in circuit-breaker oper-ti.on.

If tne insulating properties of tne gap do not

build up t c faster rate than the recovery volte acros

the gap, the Ep Will break down. Tiis will cause

resumption or re-strike of the rc with consequent high

voltage icìI current trrisient throughout trie circuit.

These mnsy prove injurious to the breaker sn associated

euipm3nt, &nd in an extreme cuse of repete re-strikes,

the circuit-breaker coui be etroye.

In ap1iCations where it is feared that tne

external circuits will cause ¿ hign rate of rise of

recovery volte, resistance (3, o.705) may be witcheI

across the gap to decrease this rate of rise of recovery

voltage. the operetion is ione either aiechÀnic&lly or

electrically by diverting the arc through a resistor.

rhe resistor is removed from the circuit a few cycles

Ï:ter by a further switoning operation.

In direct current relay circuits, it is n estab-

li.hed practice to put a resistor in sries with a capa-

citor ucross the gap. £ne;e bill absorb nd dissipate

the enery frow the iciuctive elenents of tne circuit and

reduce the magnitude f the over-voltage n.i the iurtion

of the urc.

.3

fhe purpose f this tiesis is to study the effect

of a resistor ami caoscitor in series across the aç of

ari a!ternatin curreflt circuit-oreaker. Ine recDv-ery

v1tage curves are ca1cu1ae anti copare1 for various

resist ce-cetacitnce corabthations.

L

ME2HOD OF INVESTIGATION

Rather than set up arid test an actual circuit, a

mathematical analysis was chosen for several reasons:

(1) Phe lack of facilities for accurate synchron-

izatin of the start f the break with the

phase angle of the current.

(2) On severe recovery voltage rises, an actual

breaker wouli re-strike; probably oefore

its essential rate of rise curve could be

obtained.

(3) 11he capricious behaviour of circuit-breakers

and ilelectric breakdown charcter1stics lend

tnemseïves only to statistical averages of

performance. This would lea to an

excessive number of tests, if many circuit

configurations were to be investigatei.

3ecause f the cotnílexity of tne circuit and

number of cases to be investited, the roblem was

written up as a computer proraw for the Aiwac III-E

Computer at Oregon State University.

Figure 1 is the circuit that .as invesciated.

Its configuration is a simple approximation to actual

circuits encountered in the field..

5

'3

B1 L1 L2 R2

1

ele2

(e.) c1 C2

(er) - 6

Figure 1. Ctroult with .1u1ated circuit-breaker.

6

i'he s1mu1t breaker gap 1' ropreserite.1 by R,

which i varied tri iscrete steps, li = O fDr the breker closed aria very high value (R io6 ohms) for trie

breaker ri. Ida1 operatiDri is ssum1 by aiiow1ri the break to be initiated &t current zero ni by having an

infinitely fast separation of trie oontact oint3. .L1

rc-rop henoweaa re-trike cDmplicaticns are thereby eliinte. It as also assumed tk-iat steady state con- dittone revaile before the oreaker opened.

The rorem can acooinioate restrikes by substitut- trig a third value of h Which would approximateiy reres- ent trie voltge drop acröss the arc. Its insertion would ìeperid upon the contct separation and trie recovery

voltage. ïn or1er to keep trie irivestiation eneral and

to conserve couputer tiae, it ws eoide1 not to iric..ude

restrikes, since each type f circuit-breaker :ould have

its o'n breakdown cnarcteri.3tics.

2rie sending end and receivîri end voltages are of

the form

e, =

(1) e E oos(WT + e) r rtnax

wherec = iO/rrdians/ec (syiwnroaous fre.uncy) and

is the iisplacem.ent angle of trie receiving end voltage

in radians.

the sernUng end paraaeters are li1, L1 arid C1 and

the rece1v1n end parameters are R2, L2 and C2.

rhe resistor and Gapacitor shunting of the breaker

contact points is done by E3 and C3.

The loop currents are i1 i2, i arid with e1,

e2 and e3 being the voltages across the capacitors.

The voltage across the breaker contacts is e1-e2

and the plot of its magnitude versus time is the recovery

voltage curve of the breaker.

The loop equations troni Figure 1 are:

e5 L11 + a1i1 + e1

er -L2i - R212 + e2

(2)

o =-e1+e2+e3+E3t3-R3î

o -e3 - R313 + (h3 + fl)i

The prime symbol represents the order of the

derivative anti

e1 - ft(j 13) + e1(0)

7

ft e = i (L - e2(0) (3) 2 C20j j

t

3 f(i3_i4)+e3(0) e

where e(o) is the voltage across the capacitor at time O.

e

rhe metho f sDiutiOn f the 1fferent1aJ.

equations (sets 2 anc 3) f'o11ws the Itc1assica1t (4, p.179)

rather thar the ttoperati3na1 aoproch. Phis approach

was chosen because of the ability of a nputer o hn1e

systems of linear equations ani t.he fact that there were many interme1iate stegs which ooui be checkeL Phis

fc111tatea program °iebugging'1

arid was also a check upon

the proper operation of the 000puter. The final print-out would give ali values of voltage arid current if desired, at

erbitrry intrvals of' tiLne,At. . etai1e exolanation

f the mathematical method used for the soiuin of the

ejuation3 is given in the Appendix.

Figure 2. is the flow cnart of trie computer program.

after accepting the circuit parameters and other iriput

data, the constants of the caracter1stic polynomial are

ciculated. A sub-routine using Newton's ìethod is

called in to find tne roots of this poiynthnial. ?rien

found, they are tested to see 1f tiere are one real arid

two pairs of complex roots of the form

a1, a2 jb, a3 jb3,

as this is trie only combination of roots that the program

was designed to handle. Otoerwise an alarm is given arid

the computer stops. It was assumed from the beg1nnirì

that the circuit parameters chosen would have roots of

this form.

Inputs J-.;

Alar. No

___________ VAr. \ i - "roots of

I Calculate polynomial J"\the proer I constante and enter

I

\for.1 I

root-solving sub-routine

lee

Solves system Caloulates Solves system for steady values at for steady state (before > time T1 when state (after breaker opens) i - O. breaker opens)

at time T1.

Solves Computes syst.ms tor systems for transient j higher traasient initial 'j derivatives > ourrent conditions. èf initial constants

conditions. A to E.

voltage S's at intervals of At. Calculates Pt'iiits out all l's and

constants e -e. + tr G to N.

1 155 + ttr

Figure 2. Flow diagra. of computer program.

lo

The magnitude of the currents and voltages before

the breaker opens is calculated by finding the general

steady state solution in the form

13 = 'SS max cos(WT + (le)

e =ß ss ssmax0s(1T+%e)

The time (2i) found when t1,, which is the

current through the breaker, equals zero. The partloul3r

values of all voltages and currents at this instant of

time re computed and stored.

the steady state solution after is inoresed, simulating the breaker opening, is also calculated at

time T1. the difference between the two steady state solutions of the values e1, e2, e3, t1 and

2' represents

the initial conditions of the resulting transient. The

transient is of the forn

ttr A_alt +_:2(B sin b2t + C cas b2t)

+ 3 (Dsirib3t+Ecosb3t) (5) -a t

er 06 4 E_a2t(H Slfl b2t + K 003 b2t)

-a t + 3 (M sin b3t + cos b3t)

The constants A to E and G to N are calculated from the

initial conditions and the roots of the characteristic

polynomial.

11

The steady st&te anJ trnsierit values are c1culated at irltervals of At, then added together nd

printed 'Dut.

1ie interval At may be shortened or lengthened as

desired during the print-out, and the whole print-out earl

be stopped and started at a new tiffle.

ich curve takes about one half hour of coíuçuter

tine. A.roximately half this time is spent on the

initial calcuistiorio and trie remaining time is spent on

the print-out with its asociated computations.

ere:

12

fiESULTS

£ie values chset for the test 1rcuit Df F1ure i

Li.6 ohms

L1 0.0307 henrys

C1 = 0.107 w1erofards

= 3.06 'ims

L2 = 0.0205 ohws

02 = 0.071 wicrofars

£hese are the eu1valent lumpe.i constants for ari avercie 69-Kv line (5, p.280), wtbh a length of 15 r1ies Dri the

sen1irig en s1e f te breaker ar a 1erith of 10 miles

Dfl trie rce1v1ng end side. One half the eu1valent luamped capacitance of e.cú line section was pUt aijcent t the :1rcu1t-braker. the sriding and reee1v1n ends

of the line were assum t terminate ori irìfthite buses.

Lutnpei capacitDrs at tnee eas wul have no effect and

were, tnerefore, eli.nitiated. ; sein en. volte (e5)

of 55 Kv peak-to-;;roun W:;S used.

'rue above line constants were hell for all the

oalculations except two. the quarititles varied were trie

breaker siuntin res1striee () ni caaoitarice (C-q). .1 -I

t\ snort circuit on th receiving end Df' tne line (er 0) Wb'S foun1 to give by far the most severe trans- tent. P ..relininary alculat1on showed trie maximum ap

vltge to reach only 5 Kv when load current was inter- rupted, coampared t almost 100 Kv for interrupting fault current. Tnis is because, ori short circuit, the voltage

i:j

puasor 1eds trie current phasor by approximately 9Q0

hen the interruption occurs at current ¿ero, the voitge lll be close to uiaximutn, 1eavin a maximum chars-e on the

line-to-ground caac1tors (C1 and C2) to 1nitite the

rult1rx, tr.nsient.

Fiure 3. (Cases flat to "e") shows the results f

vrythç h3 from 10,000 ohms to lo ohms, wriule holding C3

t a large value of i rticrofarad. frie curve of Case "a" shows the effect of

ic,o0o ot-im resistor. rhe resistance is so large that

lt$ effect is similar to tnt of an operi circuit in the

circult-oreaker shunt path. ; calculation with a small value of' C3 (0.01 microfarad) and a small value of

(lo ohms) was cade nd esseìtialìy the same curve aS "a"

was produced. !he si-e f C3 was too small to nave any

effect on the circuit recovery voltge transient. Curve

"a" can, therefore, be taken as a close approximation to

the response of the system with no shunt conrction around tne breaker.

£ne voltage (e3), across the shunting capacitor

(C3), 1ll follow tne gap voltage with time lag dic-

tted by the ii-C time constant of the shunting elements.

It is iuiossible for it ever to exceed the inxlmum &p

voltage.

The capacitor voltge (e3), as well as the re-

covery voltage, as printed out at each interval of

-e _-

-20 ____ ____ ____ _____ ____ ____ ____

-30. ____ ____ ____

-+0 ____ ____ ____

: :

-50rn e

4.)

o '-4

'.4

-7o ___ -

-90.-80.

_____ ____ ____ _____ ____ ____ ____ ____

-100 . ____ ____ ____ ____ ____ ____ e

u 100 200 300 4O0 500 600 700 800 900 TI.e in Mtoroeeoonds

çJJa_.__i C3- 1,M-t R3.10,000A

Qi!i_ C3 - 1,.t E - 1,00041-

au_i C3- R3-

Caes d C3 - i,f a 100 t.

U_1_1 C3 - 1,.af R3 - 10.tL

Pigure :3. Caloulated recovery voltage ourvi. Case. a to e.

15

______ _____ ______ _____ _____ _____ _____ _____ ______

:::\_________

:: ç

90

r/ J ! 100 200 300 eoo 500 600 700 800 900

TI.e t Mtoroeeoonds Q1!LL C3 - . i ,MÍ R - 100

Case E Cj - . i ,sd R - 1000 .'-.

Caee h C3 - i ,! 83 - +OO £. L1 and L2 oub1od

Cpu i C, - l,sct 3 *,00f .37,,M.f added to C

ç&ig i C3 - io6,.r R3 a

Csse k C3 a lOAtt R3 a

Pigure Li. Caloulatid roover voltage ourve. Case. t to k.

16

tlne (At) in the ca1cultjon. In Case "e" (l = lo ohms),

the curve of capacl.tcr voltage (e3) conformed almost

exactly to the recovery volte curve. s wa in-

creased, the curve of capacítor vItge followed the curve f recovery volte witn an increased tine l. In Case "c" (R3 = ¿400 ohns), the capacitor voltage rose

to the mariitude of the gap voltee curve only fter ri

interval of 1,000 microseconds.

In 'igure ¿4., eases "f" and "i" snow trie results f a capacitor of 0.1 rnicrofzrad in series ;,4th a resistor f 103 ohms and 1,000 ohms respectively. By comparison

with the curves of Cases "i" arid "b", there the shunting

capacitor (C3) is i microfarad, tne curves "f" arid "" hw the effect of a reduction in the size of the shunt-

Ing capacitor, by the increased frequency of tne recovery voltage trìnslerit. the curves ("f" arid "s") are closer

in snde ;;'rid values t.o trie limiting curve of Case "a", where the breaker Thutiting 1opearice nas no noticeable

effect.

Case represents a calculation with and

doubled. fhe resistor (E3) of ¿403 ohos end the capacitor (C3) f i microfarad re the same values as in Case "e". ihe effect of the increased inductance in the circuit is

to reduce the rate of rise of recovery voltage and to increase the peak value slightly.

17

fhe saíne result is observsi in Case "i". Lhe

i1ne-t-grDurLl capc1tor on the bus at the receiving

sUe of the breaker is inrease by 0.37 u11cr3f'ara, the

equivalent of a 225 cupacitor. The breaker shunting

values (R3 ari. C3) are the same as in Cases ' aal

The effect of' 1ncresing the breaker shurit1n

captcitnce (C3) is auproxLnateiy the saíae as incresing the induct&nce or caacitarice in tne connected circuits.

The r.te of rise of recovery voltage is reduced and the

peak recovry voltage is iracresed.

Curves 'j' and k" show the results euivalent to

short-circuiting the shunt cajacitor (3). Phis is

achieved in the computer program by setting C3 to an

extremely nigh value of microfrs, thereby a.Lnost

e1iinintin its effect fron tue circuit.

ith the effect of the capacitor thus elimin.ted,

the computer program coula not handle a value f j5

lower than about 500 ohms, because the characteristic

polynomial ciad three real roots. fhis would indicate

that the couding beteen trie two circuits on eac side

of the breaker had become so Dret tnat taere was one

transient oscillating term ineted of two.

;t comparison of Cases "j" and "k", with Cases "c'i

nd "b", shows that the effect of the one microfarJ

shuntiri capacitor is very small wian it. is in series

with a pure resistor f about 500 ohms or greater.

18

There is, owever, cns1derale aifference in the final

stey state current tnrough trie shunting brnci, with

ani without tuìe capaoitr. There wDuil be 13.8 awperes

rws in Case "b" where a i m1crfrd capacitDr is in

series witri the 1,000 ohm resistDr an 39 amperes rms in

Case "k" where tie capacltDr is essentially snort-

cireui.te. IwD calcuitions were me to ewphasi/e the

iifference between interru)t1ng lirect ¿ni alttrnatthg

current. frie current (i) through the breaker was

interrupted by 1n'erting R into tue circuit t the

rnaxirnuui current point in the cycle, thereby approximating

a 1irect-current interru:t1orì. The interrupted eurrerit

1:.:&s a load current of approxi::iite1y 100 amperes. ith

0.1 a1crofarad and 10 ohws shunt impedance, the peak

vDltae was 22 KV. 4hen the shunt resist:nce was

inorceused to 1,000 ohms, the peak recovery voltae went

up to Z50 Kv.

Lable 1. is a summary f the results. e value

dven for the rate of rise f rcDvery volte is the

sle of a stra1rit ìine, passing through trie zero point

an lying tnent to the recovery volte curve.

TABLE I

RESULTS Oi liECOVER1-VOL2AE TUDY

Peak Rate c. rec.v. reL.v.

Case -' - - .-" rise rise , Comments Kv vDits

sec

.-. i 10000 9.5 71" - - i and e- beyrñ ÏItts Df thnputer. ______ ______ ______ ______ ______ ______ e3 peak not

b i 1.000 60 cit _ reached by end _____ of run. ______

':: i 14.00

_____

6

______

)?

_____

j i 1c FU i6 167 79

: i 1..1 15 215 1'1

F .1 1ùO 95 OO 2

¿ .1 1000 75 '-'?B

- L1aadL2 C] I Ly 5 1_ ioubled in

size.

L i JO

______

7

______ ______ ______

- .37 J1 ______ to C2.

j 106 6:o

_____ _____

)?2

_____ _____

o

58 Li6 o

Curve similar - .1 1C: '6 O 1O: 10 uf with

higher peaks. ______

- _____

ni

_____ _____ _____ _____

-

_____

- urve similar tD !laft.

CONCLU3ION3

A resistr-capc1t.r shunt, across the ontct points f a cirutt-breaker, perfDrtus ¿ different funtin tri iritrruptiri itrect current than Ic es ri interrupt- trig a1cernattn current.

In tne flrecc current case, n ttewpt is nade to

break the ourrent f1owin tirough n inuctne. /

capacitor, shunting the contact p1nts, is needed t prevent the theoretically Infinite overvoltges that coula

occur.

In the alternating current case, the current through the inluctance has coirie to its riturci zero hen

the interruption is uie. fhe eriunt capacitor merely

¡nodifies the nature of trie resulting trnsIant, causes by

the storei ener'y in the ekutvalent line-to-grouna

capacitrs. In both cases, a resistor in aries with che shunt

capacitor .'iil en t introduce iaai'ia into the circuit, arì cut owri the nagnituie of' the transient oscillations of' voltage current.

The results of tris Investition will only be

qualitatively appiic.ble to an actual three-phase circuit. Two modifying factors that would have to oe consLIere re

the effe of the interction between the hises and the

effet of the 1stributei line constants.

21

Several enerzd onc1us1oris uay, however, be Irawn

abjut tie c::CtlDfl Cif res13tance-capitce ahurit.

.ntiecte acrcs ari aiterriatia currct clrcult-breaker:

(1) -cpac1.tor shuit 111 rduce the rate of rise f

recovery vc1tae. 10 pruce a noticeable effect,

however, tie cactr must be L1at1Ve1y large conpared

with th eu1va1ent 1umpe capac1tri3e cf the circuit. The peak recDvery volte, with a pure capacitor snut,

will be as nigh as, or igier than, that prJuce with

rio shunt whatsoever.

(2) a1ng resistance in series with the shunt capaci-

tance ciecreases the peak value of recovery vDltage but

increases its rate of rise. rhere appears to be an

optimum viue cf resistarie (about 100 ohns in this

circuit), where, iue to trie shape of the recovery

voltage curve, the effect of' reducing te 3eak value

of' recDvry voltage 1so causes a slight re.iuctio in

its rate Df rise.

(J) Increasing the resistice of tne circu1t-oreker

shunt ternis to ecreae the effect of tiìe cap&c1tnce.

A point is soon reache.i where the capacitance, in

:series with the resistance, has rio more effet on t'ne

transient performance than t'ne res1sttice aiDne.

Aith a shunting resistrice only, there is a value

of resistance (about LQQ ohms in this circuit), oelow

22

which the peak recovery voltage is no greater than the stey state gap voltae. This value f reaistnce also causes a reduction in the rate of rise jÍ' recovery voltae, cDmpare with the case where there i no &îunt eonriction. If' the shunting resistor is rnae very srtia1]. the current passing through it becomes a signif- leant Vrction of the tot1 current thterruted. The

breaking of this current then becomes as iajor an operation as rnakin the origthal break. This, however, is done in many cases (3, p.705) ; the resistance being ins'rte arid retnove all in one operation. (Li.) In this experimental circuit, a large etiough shunt capacitor to ooclify sufficiently the rìte of rise f recovery voltage, would have to have a value f approx- irntely 0.5 microfarad en be able to withotand a

temporary voltage peak Df twice normal line-to-ground vc,ltage. If this c,ndition coula be met by an

oriiriary &'O-Kv rms capacitor, it would have to have a

rating of 300 XV, or 900 KVA for a three-phase bank,

which, ab an approximate value of f7.00 per KVA

(2, p.235), would be :6,3oo.00. The physical size of this unit would be too large

to inclue within the circuit-breaker case, as is done

with a resistor shunt.

The tractical use of & resistor-capacitor shunt would depend upon the charQcteristics and cost f the

23

Circuit-breaker. rhe advantages of the decrease in the rate of rise of recovery voltage and reduced final steady state current through the shunt, would have to outweigh

the disadvantages of the high peak recovery voltage, external circuitry, and cost. The alternatives would

be pure resistor shunting or a higher voltage classif- ication breaker.

2L

3IBLIOJiA?HY

1. iowatson, A. 2i. The effect of' contact syrichrDnlzatlon. Amertcan Institute Dt Electrtcal Engineers TransactlDns 80:10-13. 1961.

2. JOhns3r1, A. A. Application t capacitors power systecus. In: iestinghuse Electric CorporatiD&s 1ectr1ca1 transrnissin nd uistribut1on refereice

bcok. 4th ed. ¿ast p1ttsburh, 1950. p. 233-264.

3. Kane, R. E. ani J. K. a1ker. 69-Kv cmpresse air circuit breaker for 5,000,000 KVA. American Institute Df' Electrical Engineers Transactions 74:705-710. 1955.

. Lago, iladwyn V. and Donald L. gaidelich. Transients in linear circuits. New 1.Drk, Ronald Press, 1958. 393 p.

5. McCann, . D. Begulatlon and losses of transmission lines. In: estinghouse Electric Corporation's Electrical trhn8rnission arid distribution reference book. Lth ed. East Pittsburgh, 1950. p. 265-289.

by R. F. Lawrence)

25 APPENDIX

MAI'HEMATICS OF THE COMPUAbR Pff O3RA1

The computer prgratu is written to calculate the

transient osci1itions of' ail currents and voltìges that

occur in the alternating-current circuit of' Figure i.,

when the value of resistance (Fib) is suddenly changed.

The program reduces a system of' differential

equations into systems of linear equations, which can be

salved by calling in a system-solving sub-routine.

In the following derivations, when the symbol of

the variable of current or voltage or associated constant has subscript, the symbol refers to one particular

Current or voltage in the circuit. If it hds no sub-

script, the explanation applies equally well to every

appropriate current or volt:ge variable in the circuit, arid the procedure will usually have to be aplied to each

current or voltage variable in turn.

Figure 2. gives the order in which the computer

program ¿oes through the sets f calculations. Phis is

not necessirily the order used in the following expian-

ations. rhe method of solution for the transient response

is the usual one applied to linear differential equations .ith cntant coefficients. A solution for the urrents

Df the form

1. = AESt (6)

is assumed, in which case trie higher aerivtives are: I st Il 2 st i Ase , i. = As

As the transient response is the free response of

the system in the absence of any forcing VUnCtiD, the

sending an. re3eLvin end volta,es (e3 and er') of

Figure 1, are set t zero. By 1ifferentiating the

ortgthal loop e.uations (sets 2 ad 3), the const&nt

terms e(0) are e11m1nste and the resulting syitem of

iiffererìtia1 ewtioris, mihen t.he current substitution

(i s Et) js made, becomes, in matrix form

st

=

° (7)

\\') w h e r e

'2- 1 (5 L1+6h1+- o t-- u ' ¿ ci, \ L;'

o (_s2Lz_sRz_2) ('2) (ci) =

\ (J ' (- \ (sil. ' C2, \ J

+ +! ' C2 03)

(-sa -! \ 3 i'

: o (_sR3_) (s(a3+)4)

27

Fr a non-zero solution, the ieterinant of the

matrix () rnLst jua1 zero.

[] =

Nie evalutiDn of this determinbnt gives the characteristic

polynDinial ÍThr the circuit in the fDrrn:

a6s+a5s-' as3+a2s2a1s+a0=O as +

This charcteritic polyn:mi.i .zill ive six separate

values f "s", or roots, in its solution. friis implies

that te transient current tr

will contain six terms,

s1t S2t. s-t st 1s5t s6t + + ìd +X

For the jariicular circuit confiurtion of

Figure 1. a

will always equal O. One root will then be

6 an as the nature f the tr.nsient res..onse is

such that lt aies out a the time (t) beoones very 1are,

liz" must te zero.

The rest of this iiscussion will consider only the

five remairxin terms in the transient. Nie characteristic

polynomial will now have the form

5 ¿. _) 2 a2s + a1 = o (8) a-s + a s + a + as

o 5 14. j

It was assumed., from the nature f tne circuit

parameters, that the solution f the char-cteristic

¿8

polynomial woul tave one real ani four complex roots

s1 = -a1

s2 = -a, + jb

53 = -a2 - jb9

= -a + jb

35 = -a, - Jb

The computer .rograui tests the roots to verify that they

are f this form before allowing the computations to

oritinue.

fhe trri1ent component of current can be written,

-a1t (-a2 + jb2)t (-a, - jb2)t + V

(9) (-a. + jb-)t (-a -jb)t -' +YE 3

which can be w.odified t the form

-a1t -a2t = M + (B sin b,t + C cos b2t)

-a t (10) J (D sin b3t + cos bt)

The constants A to E must be determined from the initial

Conditions existing in the circuit at the instant the

discontinuity is introduced (li onariged).

f'nen the value of R is suddenly changed at time

rl, the voltages e1, e2 and e across the caacitor and

the currents 12 through the inductor cannot change

instantaneously, because cf the nature of the circuit

elements. L'he transient initial conditions are these

values of current and voltge, minus the new stea3y

state values calculated for the charie circuit at time

= - 155(Tfl

+ etr(0) = e(ij) - e38 (ri)

29

(11)

To c&lculate the twenty coastnts A tnrouh for

the four transient currents, higher derivatives of tne

initial confl.tions must be found. The equation (10) of

the transient current is evaluated at tiwe t = G, s well

es its four h1her derivatives.

1tr A + C + E

= -a1A + b23 - a2C + b3D - a3E

= a12A + (-2a2b2)8 + (a72 - b22)C + (-2a)b.)D

a3 - and so on.

11his operation is erforned in the computer by building

up a matrix f the form

i

ci (N) 02

03

o

di

a)

i

e1

e2

e3

e

o i

fi i

'2

f) 3

t'Ll,

where

Cl =

C2 =

0.3 =

CLI. = cl1

d1=b2 e1=-a2

= ¿d1e1 e2 = (e12 - d12)

= + e1d2) e = (e1e2 - d1d2)

= 2de = (e2' -

The "f" and "g" c1uaìns re compiled identicaily

to the "d" and "e" columns, lth b3 substituted for b2

an a3 for a2.

This results in the system of euatins,

A

J i'

(\) = ;r (12)

1tr ¿V

1tr

This system of equations cari be solved for the

current constnts A to i, after first firidin the

initial values f the four transient currents arid their

higher ierivatives.

fo calculate the lriitll values of the transient

currents at tl.tnC t = O, and their higher derivatives,

the original ioop e.1uatlons () are re-written witn the

forcing voitaes (e5 arid er) set at zero.

Liijtr 1t1tr - eitr

-L21' Ri - 2tr 2 2tr e2

(13)

'313tr - R3i1tr eltr - e2tr - e3

313tr + (R3 + R4)i,

31

After substitutia the transient initial conditions

(11) of eltr, e2tr, ejtr, tltr 8 12tr in the above set

of equations (13), they can be solved for the values of

1ltr' 12tr' t3tr and

The higher derivatives of the transient currents

are calculated by first foruiing a watrix

¡L1 O O O

O -L2 O O () a

o a3-a3

) O O -R3(R3+R,)

Then, by differentiating the above set of equations (13)

and making use of the current-voltage relationships (3),

a system of equations is constructed and solved.

tltr _Riijtr_(iitr_itr)

( )

tZtr R2i2tr_32(3tr_12tr)

13tr 1('1tr3tr 2(3trt2tr)

11tr C3 ('3tr'4.tr)

(1h)

32

Successively higher erivattves of current are

calculated in this way, placing the results of one

calculation into the constants of the next.

Ihe relation (3) of the loop equations shows the

voltages to be an integrated function f the currents.

Because of the nature f sine and exponential functions,

tne transient voltages will nave the same form a's the

transient currents, le:

-a t -a2t GE ' + (fi Sin b2t + K cos b2t)

-a t (15) +e 3 (M sin bt + N cos b3t)

'the transient voltages are derived from the

transient currents by a further manipulation of the loop

equations (13),

eltr X -Lii;.tr - 11ltr

2tr L2itr + R212tr (16) e X

e3tr eltr - e2tr _ E3(i3tr _ l4tr)

By suitably combining the A, 13, C, D an E const-

ants of the four transient currents with tne circuit

parameters and roots of the charcteristic polynomial,

the G, 1-i, K, M and N constants of the three voltages may

be directly determined.

Conventional alternating-current phasor technique

is applied t'o calculate the steady state values of

current and voltage. the loop equations from Figure 1.

when written in terms of complex quantities are:

()

(i1ss\

1333) 01

is8! o /

The ber () indicates a complex quantity an

(a + jb)

I O

('i) f (O+jc)

where

a - R1

=

i

o (o+jc)

(d+ je) (0+ jt)

(0+jg) (h+jk)

o (m-i-jn)

f b -

e -(_2 +

h - R3

33

(17)

o

o

(m + jn)

(p + jq)

i C =-

1/1 i i k- --r- + - + - AI\j C2 C3

- n = -i- m--R3 "C3 q

3Ll

This conp1ex-number systecn f equations (i.?) Is

expanded thto a real-number syteìn by the following

manipulation:

a O O O -b O -C O (E'')ii

o o o o -e -f (rì)t (ae)ir\

o o i rr -C -g -k -r (ae)i355 G

o o m p 3 O-n-q o

b O C O a O O O (IaiYLiss

o e f O O 1 O O (In)i2SS

C g k n O O h u (Iw)1353 O

o o ti 'a O O n p (Dn)I5

(18)

After solving the abve system (18), t'ne conputer

program calls in a sub-rut1ne naniling eDruplex nuaibers,

tD out the currents tri polar frtn. Ibe voltages are

tbe calculated frcn trie circuit ccnstants and associated

currens.

At eaoh interval Df' time (1 nat) the steady

state values of current an voltage are calcuÏate in the

f orm

= cos(r1 flAt) + 4) (1)

Css = E35 max COS(w(F1 + nAt) +

The absolute value f the current pnasor is 'max and %

is its pha3e angle.

35

The transient values t' current nd voltae are

similarly calculated at each interval f time (nAt).

-a1(nAt) -a2(nAt)( sin b2(riAt) + C s b2(nAt)) ttr AE

-(aAt) (D sin b3(nAt) + E cos b3(nAt))

-a1(nAt) -a (flAt) etr 2

(u sth b2(nAt) + K cos b2(nt) -a(nt) + (M sin b3(nAt) + W cos b3(nt)

The steady state and transient values t current ani

voltage are added together at each interval of time (at)

to forai the total values. This process is carried on,

increasing "ri" by i eacr step, until enough paints are

obtained to plot the ìesire curves. The computer is

then manually stopped.

The prgram can be ruodified while it is running

through the computer by using the three jump switcìes

provided on tie control panel.

If the second switch is jn during trie calculation

part of the program, the constants of t}ie onaracteristic

polynomlal, the roots of the characteristic polynomial, arid trie results of all the systeu-slving o&erations will

be printed out. All the intermediate constants may then

5e checked. If tne first switch is on, the matrices of

the systems to be solved are printed out, allowing

further checks to be maie.